For the given functions f, g and h find fogoh and state the exact domain of fogoh f(x)=1/x g(x)=lnx h(x)=2x+15.

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Given the following functions:

f(x)= 1/x

g(x)= ln x

h(x)= 2x+15

We need to find the composite function: fogoh(x)

==> fogoh(x)= f(g(h(x)))

= f(g(2x+15))

= f(ln(2x+15))

= 1/ln(2x+15).

==> fogoh(x)= 1/ln(2x+15)

Now we will find the domain.

==> The domain is all real numbers except when the denominator is 0.

But we know that "ln" is always a positive number.

Also, 2x+15 must be greater than 0.

==> 2x+15>0

==> 2x >-15

==> x >-15/2

`==gt x in (-15/2, oo)` .............(1)

Now we need to include the domain of the input functions. h(x) and g(x).

==> The domain of g(x) is all real numbers greater than 0.

`==gt x in (0,oo)` .............(2)

==> The domain of h(x) is all real numbers

`==gt x in (-oo,oo).` .........(3)

Then we will find the common domain from (1) , (2) , and (3).

`==gt x in (-oo,oo)nn(0,oo)nn(-15/2, oo)`

`==gt x in (0,oo)`

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